Question

According to the Daily Racing Farm, the probability is about \(0.67\) that the favorite in a horse race will finish in the money (first, second or third place). In the next five races, what is the probability that the favorite finishes in the money.

a. exactly twice?

b. exactly four times

c. at least four times?

d. between two and four times, inclusive?

e. Determine the probability distribution of the random variable \(X\), the number of times the favorite finishes in the money in the next five races.

f. Identify the probability distribution of \(X\) as right skewed, symmetric or left skewed without consulting its probability distribution or drawing its probability histogram.

g. Draw a probability histogram for \(X\).

h. Use your answer from part (c) and definitions \(5.9\) and \(5.10\) on pages \(227\) and \(229\) respectively to obtain the mean and standard deviation of the random variable \(X\).

i. Use formula \(5.5\) on page \(239\) to obtain the mean and standard deviation of the random variable \(X\).

j. Interpret your answer for the mean in words.

Short answer

Part a. \(0.1613\)

Part b. \(0.3325\)

Part c. \(0.4675\)

Part d. \(0.8213\)

Part e.

\(x\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)
\(P(X=x)\)	\(0.0039\)	\(0.0097\)	\(0.1613\)	\(0.3275\)	\(0.3325\)	\(0.1350\)

Part f. Left skewed

Part g.

Part h. \(\mu =3.3498\)

\(\sigma =1.0518\)

Part i. \(\mu =3.35\)

\(\sigma =1.0514\)

Part j. On average, the favorite finish in the money in \(3.35\) out of \(5\) horse races.

Step-by-step solution

Part a. Step 1. Given information

In a horse race, the probability of a favorite finish in the money is \(0.67\). Consider the next five races.

Part a. Step 2. Calculation

From the given information it can conclude that the number of times the favorite finish in the money follows a binomial distribution with \(n=5\) and \(p=0.67\).

Let the random variable \(X\) is that the number of times the favorite finish in the money among five races.

The probability of exactly two times the favorite finish in the money is

\(P(X=2)=\left ( \frac{5}{2} \right )0.67^{2}(1-0.67)^{5-2}\)

\(=0.1613\)

Part b. Step 1. Explanation

The probability of exactly four times the favorite finish in the money is

\(P(X=4)=\left ( \frac{5}{4} \right )0.67^{4}(1-0.67)^{5-4}\)

\(=0.3325\)

Part c. Step 1. Explanation

The probability of at least four times the favorite finish in the money is

\(P(X\geq 4)=P(X=4)+P(X=5)\)

\(=\binom{5}{4}0.67^{4}(1-0.67)^{5-4}+\binom{5}{5}0.67^{5}(1-0.67)^{5-5}\)

\(=0.3325+0.1350\)

\(=0.4675\)

Part d. Step 1. Calculation

The probability of between two and four times inclusively the favorite finish in the money is

\(P(2\leq X\leq 4)=P(X=2)+P(X=3)+P(X=4)\)

\(=\binom{5}{2}0.67^{2}(1-0.67)^{5-2}+\binom{5}{3}0.67^{3}(1-0.67)^{5-3}+\binom{5}{4}0.67^{4}(1-0.67)^{5-4}\)

\(=0.1613+0.3275+0.3325\)

\(=0.8213\)

Part f. Step 1. Calculation

Evaluate the probabilities at \(x=0,1,2,3,4,5\)

\(P(X=0)=\binom{5}{0}0.67^{0}(1-0.67)^{5-0}\)

\(=0.0039\)

\(P(X=1)=\binom{5}{1}0.67^{1}(1-0.67)^{5-1}\)

\(=0.0397\)

\(P(X=2)=\binom{5}{2}0.67^{2}(1-0.67)^{5-2}\)

\(=0.1613\)

\(P(X=3)=\binom{5}{3}0.67^{3}(1-0.67)^{5-3}\)

\(=0.3275\)

\(P(X=4)=\binom{5}{4}0.67^{4}(1-0.67)^{5-4}\)

\(=0.3325\)

\(P(X=5)=\binom{5}{5}0.67^{5}(1-0.67)^{5-5}\)

\(=0.1350\)

Combining this information in a table, it can obtain the probability distribution:

\(x\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)
\(P(X=x)\)	\(0.0039\)	\(0.0097\)	\(0.1613\)	\(0.3275\)	\(0.3325\)	\(0.1350\)

Part f. Step 1. Calculation

The binomial distribution is

• Symmetric if \(p=0.5\),
• Skewed to left if \(p>0.5\),
• Skewed to right if \(p<0.5\).

In this case, \(p=0.67>0.5\).

Therefore the distribution is skewed to left.

Part g. Step 1. Explanation

The height of the bars are equal to the probability and the bars are centered about the values of \(X\).

Part h. Step 1. Calculation

The mean

\(\mu =\binom{0\times 0.0039+1\times 0.0397+2\times 0.1613}{3\times 0.3275+4\times 0.3325+5\times 0.1350}\)

\(=3.3498\)

The standard deviation

\(\sigma =\sqrt{0^{2}\times0.0039+1^{2}\times0.0397+2^{2}\times0.1613+3^{2}\times0.3275+4^{2}\times0.3325+5^{2}0.1350-3.3498^{2}}\)

\(=1.0518\)

Part i. Step 1. Calculation

The mean

\(\mu =5\times0.67\)

\(=3.35\)

The standard deviation

\(\sigma =\sqrt{5\times0.67\times(1-0.67)}\)

\(=1.0514\)

Note that the small difference with the results of part (h) are due to rounding error.

Part j. Step 1. Explanation

Interpretation:

On average, the favorite finish in the money in \(3.35\) out of \(5\) horse races.